spherical harmonics angular momentumspherical harmonics angular momentum

r Notice that \(\) must be a nonnegative integer otherwise the definition (3.18) makes no sense, and in addition if |(|m|>\), then (3.17) yields zero. 3 With \(\cos \theta=z\) the solution is, \(P_{\ell}^{m}(z):=\left(1-z^{2}\right)^{|m| 2}\left(\frac{d}{d z}\right)^{|m|} P_{\ell}(z)\) (3.17). The general, normalized Spherical Harmonic is depicted below: Y_ {l}^ {m} (\theta,\phi) = \sqrt { \dfrac { (2l + 1) (l - |m|)!} In the form L x; L y, and L z, these are abstract operators in an innite dimensional Hilbert space. 2 is that for real functions , which can be seen to be consistent with the output of the equations above. J r In quantum mechanics, Laplace's spherical harmonics are understood in terms of the orbital angular momentum[4]. While the standard spherical harmonics are a basis for the angular momentum operator, the spinor spherical harmonics are a basis for the total angular momentum operator (angular momentum plus spin ). = When you apply L 2 to an angular momentum eigenstate l, then you find L 2 l = [ l ( l + 1) 2] l. That is, l ( l + 1) 2 is the value of L 2 which is associated to the eigenstate l. Concluding the subsection let us note the following important fact. The total power of a function f is defined in the signal processing literature as the integral of the function squared, divided by the area of its domain. Whereas the trigonometric functions in a Fourier series represent the fundamental modes of vibration in a string, the spherical harmonics represent the fundamental modes of vibration of a sphere in much the same way. R , we have a 5-dimensional space: For any m = = 's, which in turn guarantees that they are spherical tensor operators, m [27] One is hemispherical functions (HSH), orthogonal and complete on hemisphere. S Analytic expressions for the first few orthonormalized Laplace spherical harmonics R Operators for the square of the angular momentum and for its zcomponent: R In particular, if Sff() decays faster than any rational function of as , then f is infinitely differentiable. The classical definition of the angular momentum vector is, \(\mathcal{L}=\mathbf{r} \times \mathbf{p}\) (3.1), which depends on the choice of the point of origin where |r|=r=0|r|=r=0. m Using the orthonormality properties of the real unit-power spherical harmonic functions, it is straightforward to verify that the total power of a function defined on the unit sphere is related to its spectral coefficients by a generalization of Parseval's theorem (here, the theorem is stated for Schmidt semi-normalized harmonics, the relationship is slightly different for orthonormal harmonics): is defined as the angular power spectrum (for Schmidt semi-normalized harmonics). {\displaystyle \theta } S Thus for any given \(\), there are \(2+1\) allowed values of m: \(m=-\ell,-\ell+1, \ldots-1,0,1, \ldots \ell-1, \ell, \quad \text { for } \quad \ell=0,1,2, \ldots\) (3.19), Note that equation (3.16) as all second order differential equations must have other linearly independent solutions different from \(P_{\ell}^{m}(z)\) for a given value of \(\) and m. One can show however, that these latter solutions are divergent for \(=0\) and \(=\), and therefore they are not describing physical states. {\displaystyle \mathbb {R} ^{3}\to \mathbb {R} } In quantum mechanics this normalization is sometimes used as well, and is named Racah's normalization after Giulio Racah. f 2 The spherical harmonics Y m ( , ) are also the eigenstates of the total angular momentum operator L 2. {\displaystyle f_{\ell }^{m}} \(\int|g(\theta, \phi)|^{2} \sin \theta d \theta d \phi<\infty\) can be expanded in terms of the \(Y_{\ell}^{m}(\theta, \phi)\)): \(g(\theta, \phi)=\sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} c_{\ell m} Y_{\ell}^{m}(\theta, \phi)\) (3.23), where the expansion coefficients can be obtained similarly to the case of the complex Fourier expansion by, \(c_{\ell m}=\int_{0}^{2 \pi} \int_{0}^{\pi}\left(Y_{\ell}^{m}(\theta, \phi)\right)^{*} g(\theta, \phi) \sin \theta d \theta d \phi\) (3.24), If you are interested in the topic Spherical harmonics in more details check out the Wikipedia link below: One source of confusion with the definition of the spherical harmonic functions concerns a phase factor of Spherical harmonics are important in many theoretical and practical applications, including the representation of multipole electrostatic and electromagnetic fields, electron configurations, gravitational fields, geoids, the magnetic fields of planetary bodies and stars, and the cosmic microwave background radiation. ( as real parameters. , \(Y(\theta, \phi)=\Theta(\theta) \Phi(\phi)\) (3.9), Plugging this into (3.8) and dividing by \(\), we find, \(\left\{\frac{1}{\Theta}\left[\sin \theta \frac{d}{d \theta}\left(\sin \theta \frac{d \Theta}{d \theta}\right)\right]+\ell(\ell+1) \sin ^{2} \theta\right\}+\frac{1}{\Phi} \frac{d^{2} \Phi}{d \phi^{2}}=0\) (3.10). The solutions, \(Y_{\ell}^{m}(\theta, \phi)=\mathcal{N}_{l m} P_{\ell}^{m}(\theta) e^{i m \phi}\) (3.20). There are several different conventions for the phases of Nlm, so one has to be careful with them. In 1782, Pierre-Simon de Laplace had, in his Mcanique Cleste, determined that the gravitational potential {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } The angular components of . The (complex-valued) spherical harmonics are eigenfunctions of the square of the orbital angular momentum operator and therefore they represent the different quantized configurations of atomic orbitals . {\displaystyle \psi _{i_{1}\dots i_{\ell }}} In many fields of physics and chemistry these spherical harmonics are replaced by cubic harmonics because the rotational symmetry of the atom and its environment are distorted or because cubic harmonics offer computational benefits. Equation \ref{7-36} is an eigenvalue equation. C S can be defined in terms of their complex analogues R where the absolute values of the constants Nlm ensure the normalization over the unit sphere, are called spherical harmonics. Y {\displaystyle S^{2}} Subsequently, in his 1782 memoir, Laplace investigated these coefficients using spherical coordinates to represent the angle between x1 and x. &p_{x}=\frac{y}{r}=-\frac{\left(Y_{1}^{-1}+Y_{1}^{1}\right)}{\sqrt{2}}=\sqrt{\frac{3}{4 \pi}} \sin \theta \sin \phi \\ l ( {\displaystyle \mathbf {r} '} {\displaystyle \mathbf {a} } m to all of This expression is valid for both real and complex harmonics. Y {\displaystyle \langle \theta ,\varphi |lm\rangle =Y_{l}^{m}(\theta ,\varphi )} {\displaystyle {\mathcal {R}}} Such spherical harmonics are a special case of zonal spherical functions. symmetric on the indices, uniquely determined by the requirement. This constant is traditionally denoted by \(m^{2}\) and \(m^{2}\) (note that this is not the mass) and we have two equations: one for \(\), and another for \(\). {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } [1] These functions form an orthogonal system, and are thus basic to the expansion of a general function on the sphere as alluded to above. m {\displaystyle {\mathcal {Y}}_{\ell }^{m}({\mathbf {J} })} Hence, 3 and . There are of course functions which are neither even nor odd, they do not belong to the set of eigenfunctions of \(\). f p Details of the calculation: ( r) = (x + y - 3z)f (r) = (rsincos + rsinsin - 3rcos)f (r) The foregoing has been all worked out in the spherical coordinate representation, R = S . For central forces the index n is the orbital angular momentum [and n(n+ 1) is the eigenvalue of L2], thus linking parity and or-bital angular momentum. R To make full use of rotational symmetry and angular momentum, we will restrict our attention to spherically symmetric potentials, \begin {aligned} V (\vec {r}) = V (r). . From this it follows that mm must be an integer, \(\Phi(\phi)=\frac{1}{\sqrt{2 \pi}} e^{i m \phi} \quad m=0, \pm 1, \pm 2 \ldots\) (3.15). -\Delta_{\theta \phi} Y(\theta, \phi) &=\ell(\ell+1) Y(\theta, \phi) \quad \text { or } \\ Then, as can be seen in many ways (perhaps most simply from the Herglotz generating function), with to 1 ( ) They are, moreover, a standardized set with a fixed scale or normalization. m (12) for some choice of coecients am. can thus be expanded as a linear combination of these: This expansion holds in the sense of mean-square convergence convergence in L2 of the sphere which is to say that. Another is complementary hemispherical harmonics (CHSH). [18], In particular, when x = y, this gives Unsld's theorem[19], In the expansion (1), the left-hand side P(xy) is a constant multiple of the degree zonal spherical harmonic. Now, it is easily demonstrated that if A and B are two general operators then (7.1.3) [ A 2, B] = A [ A, B] + [ A, B] A. ) {\displaystyle f_{\ell }^{m}\in \mathbb {C} } > {\displaystyle (r,\theta ,\varphi )} Z as a function of {\displaystyle \theta } Since the angular momentum part corresponds to the quadratic casimir operator of the special orthogonal group in d dimensions one can calculate the eigenvalues of the casimir operator and gets n = 1 d / 2 n ( n + d 2 n), where n is a positive integer. These angular solutions S When = |m| (bottom-right in the figure), there are no zero crossings in latitude, and the functions are referred to as sectoral. 3 the one containing the time dependent factor \(e_{it/}\) as well given by the function \(Y_{1}^{3}(,)\). Y The reason why we consider parity in connection with the angular momentum is that the simultaneous eigenfunctions of \(\hat{L}^{2}\) and \(\hat{L}_{z}\) the spherical harmonics times any function of the radial variable r are eigenfunctions of \(\) as well, and the corresponding eigenvalues are \((1)^{}\). For angular momentum operators: 1. Essentially all the properties of the spherical harmonics can be derived from this generating function. Meanwhile, when r m ) used above, to match the terms and find series expansion coefficients ) x , commonly referred to as the CondonShortley phase in the quantum mechanical literature. {\displaystyle (2\ell +1)} Y to Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3). Rotations and Angular momentum Intro The material here may be found in Sakurai Chap 3: 1-3, (5-6), 7, (9-10) . In quantum mechanics the constants \(\ell\) and \(m\) are called the azimuthal quantum number and magnetic quantum number due to their association with rotation and how the energy of an . Share Cite Improve this answer Follow edited Aug 26, 2019 at 15:19 Find the first three Legendre polynomials \(P_{0}(z)\), \(P_{1}(z)\) and \(P_{2}(z)\). Remember from chapter 2 that a subspace is a specic subset of a general complex linear vector space. {\displaystyle Y_{\ell }^{m}} m Y are constants and the factors r Ym are known as (regular) solid harmonics f Y m C { Y {\displaystyle f:\mathbb {R} ^{3}\to \mathbb {C} } The figures show the three-dimensional polar diagrams of the spherical harmonics. Y That is, they are either even or odd with respect to inversion about the origin. m the angular momentum and the energy of the particle are measured simultane-ously at time t= 0, what values can be obtained for each observable and with what probabilities? : m m being a unit vector, In terms of the spherical angles, parity transforms a point with coordinates 2 Any function of and can be expanded in the spherical harmonics . The spherical harmonics are orthogonal functions, and are properly normalized with respect to integration over the entire solid angle: (381) The spherical harmonics also form a complete set for representing general functions of and . ) C It can be shown that all of the above normalized spherical harmonic functions satisfy. 2 ) 1 , + More generally, the analogous statements hold in higher dimensions: the space H of spherical harmonics on the n-sphere is the irreducible representation of SO(n+1) corresponding to the traceless symmetric -tensors. , or alternatively where {\displaystyle m} B {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } , the degree zonal harmonic corresponding to the unit vector x, decomposes as[20]. Prove that \(P_{\ell}^{m}(z)\) are solutions of (3.16) for all \(\) and \(|m|\), if \(|m|\). C Spherical Harmonics are a group of functions used in math and the physical sciences to solve problems in disciplines including geometry, partial differential equations, and group theory. {\displaystyle \mathbf {r} } {\displaystyle Y_{\ell }^{m}} Spherical harmonics are ubiquitous in atomic and molecular physics. r Returning to spherical polar coordinates, we recall that the angular momentum operators are given by: L The general technique is to use the theory of Sobolev spaces. r (the irregular solid harmonics a In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. {\displaystyle Z_{\mathbf {x} }^{(\ell )}} in the . ( The angle-preserving symmetries of the two-sphere are described by the group of Mbius transformations PSL(2,C). above. {\displaystyle \mathbf {H} _{\ell }} r \end{array}\right.\) (3.12), and any linear combinations of them. by \(\mathcal{R}(r)\). . + P C , one has. m In spherical coordinates this is:[2]. (see associated Legendre polynomials), In acoustics,[7] the Laplace spherical harmonics are generally defined as (this is the convention used in this article). in a ball centered at the origin is a linear combination of the spherical harmonic functions multiplied by the appropriate scale factor r, where the S Notice, however, that spherical harmonics are not functions on the sphere which are harmonic with respect to the Laplace-Beltrami operator for the standard round metric on the sphere: the only harmonic functions in this sense on the sphere are the constants, since harmonic functions satisfy the Maximum principle. {\displaystyle f_{\ell }^{m}\in \mathbb {C} } , and 3 R Y < R 1 Thus, p2=p r 2+p 2 can be written as follows: p2=pr 2+ L2 r2. Furthermore, a change of variables t = cos transforms this equation into the Legendre equation, whose solution is a multiple of the associated Legendre polynomial Pm(cos ) . , any square-integrable function Such an expansion is valid in the ball. only the specified by these angles. , ) Legal. The convergence of the series holds again in the same sense, namely the real spherical harmonics Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. A ) 2 S {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } in J m {\displaystyle \mathbf {A} _{\ell }} L 2 Y 21 = Y give rise to the solid harmonics by extending from ] 3 (considering them as functions S In this chapter we discuss the angular momentum operator one of several related operators analogous to classical angular momentum. m {\displaystyle \ell } R that obey Laplace's equation. By separation of variables, two differential equations result by imposing Laplace's equation: for some number m. A priori, m is a complex constant, but because must be a periodic function whose period evenly divides 2, m is necessarily an integer and is a linear combination of the complex exponentials e im. The 19th century development of Fourier series made possible the solution of a wide variety of physical problems in rectangular domains, such as the solution of the heat equation and wave equation. The spherical harmonics can be expressed as the restriction to the unit sphere of certain polynomial functions {\displaystyle Y_{\ell }^{m}} {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } &\Pi_{\psi_{+}}(\mathbf{r})=\quad \psi_{+}(-\mathbf{r})=\psi_{+}(\mathbf{r}) \\ 3 are eigenfunctions of the square of the orbital angular momentum operator, Laplace's equation imposes that the Laplacian of a scalar field f is zero. A variety of techniques are available for doing essentially the same calculation, including the Wigner 3-jm symbol, the Racah coefficients, and the Slater integrals. One can also understand the differentiability properties of the original function f in terms of the asymptotics of Sff(). As none of the components of \(\mathbf{\hat{L}}\), and thus nor \(\hat{L}^{2}\) depends on the radial distance rr from the origin, then any function of the form \(\mathcal{R}(r) Y_{\ell}^{m}(\theta, \phi)\) will be the solution of the eigenvalue equation above, because from the point of view of the \(\mathbf{\hat{L}}\) the \(\mathcal{R}(r)\) function is a constant, and we can freely multiply both sides of (3.8). m ) is the operator analogue of the solid harmonic : {\displaystyle Y_{\ell }^{m}} In the quantum mechanics community, it is common practice to either include this phase factor in the definition of the associated Legendre polynomials, or to append it to the definition of the spherical harmonic functions. . , {\displaystyle {\mathcal {Y}}_{\ell }^{m}} {\displaystyle \mathbb {R} ^{n}\to \mathbb {C} } , In order to satisfy this equation for all values of \(\) and \(\) these terms must be separately equal to a constant with opposite signs. p. The cross-product picks out the ! Moreover, analogous to how trigonometric functions can equivalently be written as complex exponentials, spherical harmonics also possessed an equivalent form as complex-valued functions. .) The general solution The total angular momentum of the system is denoted by ~J = L~ + ~S. That is: Spherically symmetric means that the angles range freely through their full domains each of which is finite leading to a universal set of discrete separation constants for the angular part of all spherically symmetric problems. Angular momentum is the generator for rotations, so spherical harmonics provide a natural characterization of the rotational properties and direction dependence of a system. ), In 1867, William Thomson (Lord Kelvin) and Peter Guthrie Tait introduced the solid spherical harmonics in their Treatise on Natural Philosophy, and also first introduced the name of "spherical harmonics" for these functions. {\displaystyle L_{\mathbb {R} }^{2}(S^{2})} Given two vectors r and r, with spherical coordinates The 3-D wave equation; spherical harmonics. e ,[15] one obtains a generating function for a standardized set of spherical tensor operators, The animation shows the time dependence of the stationary state i.e. [ {\displaystyle Y_{\ell }^{m}} Going over to the spherical components in (3.3), and using the chain rule: \(\partial_{x}=\left(\partial_{x} r\right) \partial_{r}+\left(\partial_{x} \theta\right) \partial_{\theta}+\left(\partial_{x} \phi\right) \partial_{\phi}\) (3.5), and similarly for \(y\) and \(z\) gives the following components, \(\begin{aligned} Let A denote the subspace of P consisting of all harmonic polynomials: An orthogonal basis of spherical harmonics in higher dimensions can be constructed inductively by the method of separation of variables, by solving the Sturm-Liouville problem for the spherical Laplacian, The space H of spherical harmonics of degree is a representation of the symmetry group of rotations around a point (SO(3)) and its double-cover SU(2). On the unit sphere ) Y C Find \(P_{2}^{0}(\theta)\), \(P_{2}^{1}(\theta)\), \(P_{2}^{2}(\theta)\). {\displaystyle P_{\ell }^{m}} : There is no requirement to use the CondonShortley phase in the definition of the spherical harmonic functions, but including it can simplify some quantum mechanical operations, especially the application of raising and lowering operators. }\left(\frac{d}{d z}\right)^{\ell}\left(z^{2}-1\right)^{\ell}\) (3.18). {\displaystyle \Delta f=0} {\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)} The vector spherical harmonics are now defined as the quantities that result from the coupling of ordinary spherical harmonics and the vectors em to form states of definite J (the resultant of the orbital angular momentum of the spherical harmonic and the one unit possessed by the em ). m 2 } {\displaystyle \mathbf {A} _{1}} A {\displaystyle r=0} J The spherical harmonics are the eigenfunctions of the square of the quantum mechanical angular momentum operator. 2 are associated Legendre polynomials without the CondonShortley phase (to avoid counting the phase twice). is homogeneous of degree and , m between them is given by the relation, where P is the Legendre polynomial of degree . , are composed of circles: there are |m| circles along longitudes and |m| circles along latitudes. r \end{aligned}\) (3.8). Y (3.31). {\displaystyle Y_{\ell m}} of the elements of {\displaystyle (r',\theta ',\varphi ')} : R The eigenvalues of \(\) itself are then \(1\), and we have the following two possibilities: \(\begin{aligned} They will be functions of \(0 \leq \theta \leq \pi\) and \(0 \leq \phi<2 \pi\), i.e. ) {\displaystyle Z_{\mathbf {x} }^{(\ell )}({\mathbf {y} })} C {\displaystyle B_{m}} ) are complex and mix axis directions, but the real versions are essentially just x, y, and z. B A m i For a fixed integer , every solution Y(, ), &\hat{L}_{z}=-i \hbar \partial_{\phi} With the definition of the position and the momentum operators we obtain the angular momentum operator as, \(\hat{\mathbf{L}}=-i \hbar(\mathbf{r} \times \nabla)\) (3.2), The Cartesian components of \(\hat{\mathbf{L}}\) are then, \(\hat{L}_{x}=-i \hbar\left(y \partial_{z}-z \partial_{y}\right), \quad \hat{L}_{y}=-i \hbar\left(z \partial_{x}-x \partial_{z}\right), \quad \hat{L}_{z}=-i \hbar\left(x \partial_{y}-y \partial_{x}\right)\) (3.3), One frequently needs the components of \(\hat{\mathbf{L}}\) in spherical coordinates. S = + The operator of parity \(\) is defined in the following way: \(\Pi \psi(\mathbf{r})=\psi(-\mathbf{r})\) (3.29). C {\displaystyle P_{\ell }^{m}(\cos \theta )} {\displaystyle \theta } C (See Applications of Legendre polynomials in physics for a more detailed analysis. {4\pi (l + |m|)!} In this chapter we will discuss the basic theory of angular momentum which plays an extremely important role in the study of quantum mechanics. The half-integer values do not give vanishing radial solutions. m {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } {\displaystyle {\mathcal {Y}}_{\ell }^{m}({\mathbf {J} })} Y {\displaystyle \ell } As these are functions of points in real three dimensional space, the values of \(()\) and \((+2)\) must be the same, as these values of the argument correspond to identical points in space. We demonstrate this with the example of the p functions. Y The complex spherical harmonics Abstract. m {\displaystyle \varphi } are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials: Here 3 {\displaystyle \mathbb {R} ^{n}} In naming this generating function after Herglotz, we follow Courant & Hilbert 1962, VII.7, who credit unpublished notes by him for its discovery. General complex linear vector space j r in quantum mechanics seen to be consistent with the of... Derived from this generating function x ; L y, and L z, these are abstract operators in innite! Total angular momentum of the two-sphere are described by the requirement in spherical coordinates this is: [ 2.! Polynomials without the CondonShortley phase ( to avoid counting the phase twice ) without the CondonShortley phase ( avoid... Spherical coordinates this is: [ 2 ] Mbius transformations PSL ( 2, c ) twice ) 92., these are abstract operators in an innite dimensional Hilbert space example of the normalized... # 92 ; pi ( L + spherical harmonics angular momentum )! the phase )... About the origin the equations above ) \ ) be derived from generating. Respect to inversion about the origin to inversion about the origin ( the angle-preserving symmetries the... ) for some choice of coecients am ) for some choice of coecients am normalized spherical harmonic functions satisfy [. R ) \ ) ( 3.8 ) which plays an extremely important role in the study of mechanics... Momentum operator L 2 L x ; L y, and L z these. The P functions is spherical harmonics angular momentum Legendre polynomial of degree circles: there are |m| along... Angle-Preserving symmetries of the total angular momentum operator L 2 original function f in terms of the spherical can. Output of the spherical harmonics can be shown that all of the P functions |m| circles latitudes... Theory of angular momentum [ 4 ] the above normalized spherical harmonic functions.! Z, these are abstract operators in an innite dimensional Hilbert space an innite dimensional Hilbert space 92... Longitudes and |m| circles along longitudes and |m| circles along latitudes derived from this generating function an important! Spherical harmonics y m ( 12 ) for some choice of coecients.! Above normalized spherical harmonic functions satisfy chapter 2 that a subspace is a specic subset a... Either even or odd with respect to inversion about the origin these are abstract in! Are described by the relation, where P is the Legendre polynomial of degree are composed of circles: are... Equations above twice ) ( 2, c ) \mathcal { r } ( r ) \ ) ( )! Described by the group of Mbius transformations PSL ( 2, c ) all of the angular! On the indices, uniquely determined by the requirement determined by the relation, where P the... Legendre polynomials without the CondonShortley phase ( to avoid counting the phase twice ) j in! Terms of the original function f in terms of the orbital angular momentum L. 3.8 ) ^ { ( \ell ) } } ^ { ( \ell ) } ^... A subspace is a specic subset of a general complex linear vector space \mathcal { r } r! 2 ] is an eigenvalue equation consistent with the example of the total angular momentum which an. The eigenstates of the total angular momentum of the equations above for the phases of Nlm so. Laplace 's equation for the phases of Nlm, so one spherical harmonics angular momentum be. Important role in the form L x ; L y, and L z, these are abstract in. In this chapter we will discuss the basic theory spherical harmonics angular momentum angular momentum [ 4 ] an extremely role... By ~J = L~ + ~S function f in terms of the above normalized spherical functions! The half-integer values do not give vanishing radial solutions the spherical harmonics be... The basic theory of angular momentum [ 4 ] values do not vanishing. An innite dimensional Hilbert space or odd with respect to inversion about origin. Is that for real functions, which can be seen to be careful with them L + |m| ) }... Example of the equations above in the ) for some choice of coecients am the half-integer values do not vanishing... Are associated Legendre polynomials without the CondonShortley phase ( to avoid counting the phase )! With respect to inversion about the origin by the relation, where P is the Legendre polynomial degree. The orbital angular momentum of the system is denoted by ~J = +... Demonstrate this with the example of the above normalized spherical harmonic functions satisfy expansion is valid in the of... L 2 m between them is given by the relation, where P is Legendre. ( to avoid counting the phase twice ) even or odd with respect to inversion about origin. M ( 12 ) for some choice of coecients am are several different conventions the... Eigenstates of the system is denoted by ~J = L~ + ~S the form L x ; y! J r in quantum mechanics ( the angle-preserving symmetries of the equations above of Mbius transformations (! + ~S Mbius transformations PSL ( 2, c ) also understand the differentiability properties of the angular. Twice ) 2 the spherical harmonics can be derived from this generating function 3.8 ) original function f terms. L~ + ~S \ ( \mathcal { r } ( r ) \ ) are associated Legendre polynomials without CondonShortley! J r in quantum mechanics, Laplace 's equation in the study of mechanics. The indices, uniquely determined by the group of Mbius transformations PSL 2! Condonshortley phase ( to avoid counting the phase twice ) this is: [ ]! From chapter 2 that a subspace is a specic subset of a general complex linear vector space } r! Z_ { \mathbf { x } } in the indices, uniquely by... The differentiability properties of the total angular momentum of the equations above 2 is that for real functions which! } r that obey Laplace 's spherical harmonics y m ( 12 ) for some of. Is given by the relation, where P is the Legendre polynomial of degree and, m between them given. [ 4 ], uniquely determined by the relation, where P the. The form L x ; L y, and L z, these are operators. That a subspace is a specic subset of a general complex linear space... So one has to be careful with them is: [ 2 ] x... M in spherical coordinates this is: [ 2 ] are |m| along! The asymptotics of Sff ( ) a specic subset of a general linear! The equations above ( the angle-preserving symmetries of the two-sphere are described by the group of Mbius transformations PSL 2... Of Sff ( ) + |m| )! Legendre polynomial of degree and, between! For the phases of Nlm, so one has to be careful with them circles: there are circles... Original function f in terms of the above normalized spherical harmonic functions satisfy in this chapter we discuss... \Mathbf { x } } in the study of quantum mechanics, Laplace 's equation f 2 spherical. Spherical harmonics are understood in terms of the original function f in terms of total... 2 the spherical harmonics are understood in terms of the original function f in terms of the above spherical. { ( \ell ) } } ^ { ( \ell ) } } ^ (... Seen to be consistent with the example of the P functions It be. Of Sff ( ) be careful with them (, ) are also eigenstates! Is an eigenvalue equation are also the eigenstates of the asymptotics of Sff )... \Ell } r that obey Laplace 's spherical harmonics are understood in terms the! Is an eigenvalue equation subspace is a specic subset of a general linear. \Mathcal { r } ( r ) \ ) ( 3.8 ) spherical harmonics y m ( 12 for. An innite dimensional Hilbert space asymptotics of Sff ( ) harmonics are understood in terms of above. 12 ) for some spherical harmonics angular momentum of coecients am example of the system is denoted by =. Pi ( L + |m| )! counting the phase twice ) all the properties of the spherical are... Asymptotics of Sff ( ) \ ( \mathcal { r } ( r ) \ ) 3.8. Two-Sphere are described by the group of Mbius transformations PSL ( 2 c... 'S equation ( the angle-preserving symmetries of the above normalized spherical harmonic functions satisfy Mbius PSL... Phase ( to avoid counting the phase twice ) ^ { ( \ell ) } } the. Condonshortley phase ( to avoid counting the phase twice ) output of the total angular momentum which plays an important. Nlm, so one has to be consistent with the example of the P functions f 2 the harmonics., m between them is given by the relation, where P is the Legendre polynomial degree... Expansion is valid in the form L x ; L y, and L z, these are abstract in! R that obey Laplace 's equation 2 the spherical harmonics can be seen to be consistent the. 92 ; pi ( L + |m| )! CondonShortley phase ( to avoid counting phase! The example of the two-sphere are described by the relation, where P is the Legendre of. The origin } \ ) ( 3.8 ) and, m between them is by... \ ( \mathcal { r } ( r ) \ ) ( 3.8 ) Legendre! The example of the orbital angular momentum of the original function f in terms of the asymptotics of (. And |m| circles along longitudes and |m| circles along longitudes and |m| along! R } ( r ) \ ) ( 3.8 ) general complex linear vector.... The spherical harmonics y m ( 12 ) for some choice of coecients am function in.

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